Simplify the following expression and state the condition under which the simplification is valid. $t = \dfrac{-8z^2 - 64z - 56}{z^3 + 5z^2 - 14z}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ t = \dfrac {-8(z^2 + 8z + 7)} {z(z^2 + 5z - 14)} $ $ t = -\dfrac{8}{z} \cdot \dfrac{z^2 + 8z + 7}{z^2 + 5z - 14} $ Next factor the numerator and denominator. $ t = - \dfrac{8}{z} \cdot \dfrac{(z + 7)(z + 1)}{(z + 7)(z - 2)}$ Assuming $z \neq -7$ , we can cancel the $z + 7$ $ t = - \dfrac{8}{z} \cdot \dfrac{z + 1}{z - 2}$ Therefore: $ t = \dfrac{ -8(z + 1)}{ z(z - 2)}$, $z \neq -7$